5.1 Simplify the following expression to ONE trigonometric term: $$\frac{\sin x}{\cos x \cdot \tan x} + \frac{\sin(180^\circ + x) \cdot \cos(90^\circ - x)}{\cos x \cdot \tan x}$$ 5.2 Without using a calculator, determine the value of: $$\frac{\sin 2^\circ 35^\circ - \cos 2^\circ 35^\circ}{4 \sin 10^\circ \cos 10^\circ}$$ 5.3 Given: $\cos 26^\circ = m$ Without using a calculator, determine $2 \sin 77^\circ$ in terms of $m$ - NSC Mathematics - Question 5 - 2019 - Paper 2

We want to evaluate:

$\frac{\sin 2^\circ 35^\circ - \cos 2^\circ 35^\circ}{4 \sin 10^\circ \cos 10^\circ}$

Using angle identities, we can express this as:

$= \frac{- (\cos 2^\circ 35^\circ - 1)}{\cos 20^\circ}$

With further simplification:

$= \frac{- \cos 2^\circ 35^\circ + 1}{\cos 20^\circ} = \frac{2 \sin^2 20^\circ}{\cos 20^\circ} = 2 \sin 20^\circ$

To find $2 \sin 77^\circ$, we use the identity:

$\sin(90^\circ - x) = \cos x$

Thus, we have:

$2 \sin 77^\circ = 2 \cos 13^\circ$

Next, we express $\cos 13^\circ$ in terms of $m$. Since:

$\cos 26^\circ = m$

We can derive:

$\cos 13^\circ = \sqrt{1 - \sin^2 13^\circ}$

Hence, we conclude:

$2 \sin 77^\circ = 2 \sqrt{1 - (1 - m^2)} = 2m$

For the function:

$f(x) = \sin(x + 25^\circ)\cos(\sin(25^\circ) - \cos(x + 25^\circ)\sin 15^\circ)$

We can set:

$\sin(x + 100^\circ) = \tan 165^\circ$

Now, using the general solution for sine:

$x + 100^\circ = 195.54^\circ + k\cdot 360^\circ\, or \quad x + 100^\circ = 344.46^\circ + k\cdot 360^\circ$

Solving these leads to:

$x = 195.54^\circ - 100^\circ + k\cdot 360^\circ\, or\, x = 344.46^\circ - 100^\circ + k\cdot 360^\circ$

The minimum occurs when:

$f(x) = \sin(x + 100^\circ)$

From the previous calculations, we know that:

$x + 100^\circ = 270^\circ\, or \quad x + 100^\circ = 260^\circ$

Thus solving gives:

$x = 170^\circ\, or \quad x = 160^\circ$